Cooling by Time Reversal of Atomic Matter Waves
J. Martin, B. Georgeot, and D. L. ShepelyanskyLaboratoire de Physique The´orique, Universite´ de Toulouse III, CNRS, 31062 Toulouse, France (Received 25 October 2007; published 1 February 2008)
We propose an experimental scheme which allows us to realized approximate time reversal of matter waves for ultracold atoms in the regime of quantum chaos. We show that a significant fraction of the atoms return back to their original state, being at the same time cooled down by several orders of magnitude. We give a theoretical description of this effect supported by extensive numerical simulations. The proposed scheme can be implemented with existing experimental setups.
DOI:10.1103/PhysRevLett.100.044106 PACS numbers: 05.45.Mt, 03.75.b, 37.10.De, 37.10.Vz
The statistical theory of gases developed by Boltzmann leads to macroscopic irreversibility and entropy growth even if dynamical equations of motion are time reversible. This contradiction was pointed out by Loschmidt and is now known as the Loschmidt paradox [1]. The reply of Boltzmann relied on the technical difficulty of velocity reversal for material particles [2]: a story tells that he simply said ‘‘then go and do it’’. The modern resolution of this famous dispute came with the development of the theory of dynamical chaos [3–5]. Indeed, for chaotic dy-namics small perturbations grow exponentially with time, making the motion practically irreversible. This explana-tion is valid for classical dynamics, while the case of quan-tum dynamics requires special consideration. Indeed, in the quantum case the exponential growth takes place only during the rather short Ehrenfest time [6], and the quantum evolution remains stable and reversible in presence of small perturbations [7]. Quantum reversibility in presence of various perturbations has been actively studied in recent years and is now described through the Loschmidt echo (see [8] and references therein). This quantity measures the effect of perturbations and is characterized by the fidelity
ftr jh p2trj 0ij2, where j pi is the time reversed
wave function in presence of perturbations, j i is the unperturbed one, and tr is the moment of time reversal. Experimental implementations of time reversibility for quantum dynamics or propagating waves have been real-ized with spin systems (spin echo technique) [9], acoustic [10] and electromagnetic [11] waves, resulting in various technological applications. Surprisingly enough, the re-versibility signal becomes stronger and more robust in the case of chaotic ray dynamics [10]. However, despite the significant experimental progress made recently in the control of quantum systems, the time reversal of matter waves has not been performed so far.
Here we present a concrete experimental proposal of an effective time reversal of atomic matter waves. The pro-posal relies on the kicked rotator model, which is a corner-stone model of quantum chaos [6,7,12]. This model has been built up experimentally with cold atoms in kicked optical lattices [13–16]. Recent progress allowed to
imple-ment this model with ultracold atoms and Bose-Einstein condensates (BEC) [17–20], with high-precision subrecoil definition of the momentum of the atoms, allowing, for example, to observe [17,19] high order quantum reso-nances [21]. We show that these experimental techniques allow to perform time reversal for a significant part of the atoms. Surprisingly, this fraction of the atoms becomes cooled down by several orders of magnitude during the process. We call this new cooling mechanism Loschmidt
cooling since it is directly related to the time reversibility.
The quantum kicked rotator corresponds to the quanti-zation of the Chirikov standard map [4,5]:
p p k sinx; x x T p; (1) where x is the position and p the momentum of an atom, and bars denote the variables after one map iteration. Here
xis a continuous variable in the interval 1; 1. The physical process behind corresponds to rapid change of momentum created by a kick of optical lattice followed by a free propagation of the atoms between periodic kicks of period T. The classical dynamics depends only on the single parameter K kT and undergoes a transition from integrability to chaos when K is increased. Global chaos sets in for K > Kc 0:9716 . . . . The dynamics of (1) is time reversible, e.g., by inverting all velocities at the middle of the free motion between two kicks.
The quantum evolution over one period is described by a unitary operator ^Uacting on a wave function:
^U eiT ^p2=2
eik cos ^x ; (2) where the momentum p is measured in recoil units, and T plays the role of an effective Planck constant. The momen-tum operator ^p i@=@x has eigenvalues p n , where n is an integer and is the quasimomentum for a wave propagating in the x direction. The particle energy is
E Erp2=2, where Er is the recoil energy. We consider
noninteracting atoms. It is convenient to express the time t in number of map iterations. In experiments, values of time up to t 150 have been achieved [16]. Also, a very narrow initial momentum distribution down to rms 0:002 can be reached with BEC [17]. Values as high as k 4 PRL 100, 044106 (2008) P H Y S I C A L R E V I E W L E T T E R S 1 FEBRUARY 2008week ending
have been realized experimentally with T varying between 1 and 4 [16].
To perform the time reversal, we first write T as T 4 . After triterations of (2), we interchange the order of kick and free propagation, change T to T0 4 and
k to k, and let the system evolve during another tr iterations. Such a modification of T can be easily realized experimentally (see, e.g., [17] ). The sign of k can be inverted by changing the sign of the detuning between the laser and the atomic transition frequencies or through a shift of the optical lattice by half a wavelength. Then for
t > trthe map (2) becomes ^Ur eik cos ^xeiT0p^2=2
where the second operator acts on momentum eigenstates jn i as
eiT0n2=2 eiTn2=2
e8in=2. Thus U^r
^
Uye8i ^n=2 and the components with 0 (integer values of p) are exactly reversed, while for other small values the time reversal works only approximately. This determines the meaning of approximate time reversal. Another way of inverting k is to use in the first and last propagation steps for t > trthe value T00 2 instead of T0. Indeed, for 0 we have ein^2
eik cos ^xein^2
eik cos ^x, and thus inversion of k works again exactly for
0 and approximately for small values. In the following, we use the first method of k inversion, but we checked that the second method gives similar results.
To characterize the wave packet dynamics, we use the inverse participation ratio (IPR) defined by t P
pWpt=PpWp2t, where Wpt jhpj tij2 is the
probability in momentum space at time t. Another useful quantity is the probability at zero momentum Wt W0t. For numerical simulations we used up to N 223 discre-tized momentum states with p 1=40 000. Both quan-tities tand Wtare rescaled by their values at t 0 that makes them independent of p. Their dependence on time is shown in Fig.1which clearly demonstrates time reversal of both quantities. The return for tis not perfect due to the contribution of nonzero quasimomentum components,
whereas the curve for Wtis perfectly symmetric. We note that t shows a diffusive growth for t < 4 followed by a saturation due to dynamical localization [6,7,12,13] for
t tr.
The momentum probability distributions at initial t 0 and final t tf 2tr times are shown in Fig. 2. The
striking feature is that the final distribution in 0:5 < p < 0:5 is much more narrow than the initial one and has the same maximal value since the probability at 0 returns exactly. The shape of the reversed peak is independent of
T0. This can be interpreted as a cooling of the atoms which remained in this momentum range that defines the
Loschmidt cooling mechanism. The narrowing of the
cen-tral peak means that in compensation a significant fraction of atoms has obtained higher momentum values p as is clearly seen in the right inset of Fig.2. But even if the full distribution is rather broad the reversed peak is clearly dominant. On the contrary, the left inset showing the distribution at t trdisplays homogeneous chaotic distri-bution of momentum components. Thus it is the time reversal which produces the peak at the origin and per-forms effective cooling. It is natural to define the size of the return Loschmidt peak by its half width L with
WL2tr W02tr=2. The fraction of returned atoms
is P P2L2LW2tr and their temperature is Tf
P2L
2L2W2tr=2P. Similarly to the case of chaotic
acoustic cavities [10], quantum chaos makes the time reversed peak more visible. From an experimental view-point, the atoms outside the reversed peak can be elimi-nated by an escape process while those inside can be kept by switching on a suitable trap potential or attraction between atoms (e.g., Feschbach resonance). Such a proce-dure is similar to the process of evaporative cooling.
FIG. 1 (color online). Dependence on time t of the rescaled IPR t=0 (solid curve with blue/black squares) and of the
rescaled inverse probability at zero momentum W0=Wt(dashed
curve with red/gray dots). Initial state is a Gaussian wave packet with rms 0:002 and kBT0=Er 2=2 2 106 (see
Fig. 2). Here k 4:5, T 4 for 0 t tr, and T0
4 for tr< t 2tr with 2 and tr 10.
FIG. 2 (color online). Initial states probability (dashed curves) and final return probability (solid curves) shown as a function of quasimomentum . Here kBT0=Er 2 104 (red/gray
curves) and kBT0=Er 2 106 (blue/black curves), the latter
corresponding to data of Fig.1and being similar to experimental conditions [17]. For return probabilities, the central peaks coin-cide for the two values of T0. Insets: probability Wpat t tr
10 (left) and final probability Wp at t tf 2tr (right) on a
larger scale with p n for kBT0=Er 2 106. All
dis-tributions are scaled by the value of the initial probability at zero momentum ( 0). Parameters are as in Fig.1.
PRL 100, 044106 (2008) P H Y S I C A L R E V I E W L E T T E R S 1 FEBRUARY 2008week ending
However, in our case the effective evaporation takes place very rapidly due to dynamical chaos.
The variation of the final return probability distribution in energy E Erp2=2 with k is shown in Fig. 3. When k increases the distribution in energy becomes more and more narrow, so that Loschmidt cooling becomes more efficient. This is related to the fact that the dynamics becomes more chaotic as k increases. The temperature Tf
drops by almost 2 orders of magnitude, showing significant oscillations with k.
The decrease of Tf=T0with k is shown in more detail in Fig.4. It is related to the increase of the localization length
lof quasienergy eigenstates with k. Indeed, it is known that
l Dq=2 where Dq k2gK
q=2 is the quantum diffusion
rate, Kq 2k sinT=2 being the quantum chaos parameter [6,22]. The function gKq takes into account the effects
of quantum correlations and is given [22] by gKq 0 for Kq< Kc, gKq 0:42Kq Kc3=K2qfor Kc Kq<
4:5, and gKq 1 2J2Kq 2J21Kq 2J22Kq
2J2
3Kq for Kq 4:5 where Jm are Bessel functions.
Because of this localization there is always a residual probability W 1=l in the interval 0:5 < p < 0:5,
even in absence of time reversal. However, this residual probability is much smaller than the maximum of the reversed peak W0 1=. The width of this peak can be
estimated as follows: > 0 in ^Ur acts as a small
pertur-bation of the exactly reversible operator, whose eigenstates have M l components. This perturbation gives after time
tr an accumulated quantum phase shift in the wave func-tion 8ntr 8Mtr. Thus only the atoms with
L 1=8Mtr return to their initial state, and their
fraction is P L=. The Loschmidt temperature of
these atoms is kBTL Er2
L=2 Er=1282CD2qt2r 1
where C is a numerical constant which according to our
data is C 0:4. Thus the ratio Tf=T0 is
Tf=T0 TL=T0 TL; kBTL Er=500D2qt2r 1:
(3)
The formula (3) interpolates between the weakly perturba-tive regime with l 1 and the strong chaos regime with
l 1. This theory assumes that kBT0 Er and that the localization time scale t Dqis shorter than tr which is approximately satisfied for most k values in Fig. 4. The theory (3) satisfactorily describes the global behavior of
Tf=T0 as can be seen in Fig. 4. Small scale oscillations should be attributed to mesoscopic fluctuations. These fluctuations are stronger when is varied (see inset of Fig. 4), that should be attributed to high order quantum resonances [21] at rational values of T=4 [23]. We checked that the cooling remains robust even in presence of 1% fluctuations of during map iterations. For the case
tr t one should use M pDqtr instead of M Dq
since the diffusion takes place during the whole time interval tr. In this case, kBTL Er=1282CDqt3r 1.
It is important to evaluate the fraction P of returning atoms. The estimates given above lead to P L=/
1=Dqtr and thus:
Tf=T0 P2; (4)
that is well confirmed by the data in Fig.4. The formula (4) is written for dimension d 1. For higher dimensions it generalizes to Tf=T0 P2=d . The data displayed in Fig.5 show that P drops approximately as 1=tr, in agreement
with the estimate above. For example, the cooling by a
FIG. 3 (color online). Density plot of the return probability distribution Wp2tr as a function of the rescaled atom energy
E=kBT0 and of the kick strength k, where E Erp2=2 and
kBT0=Er 2 106; here 2 and tr 10. Colors denote
density from white (minimal) at the right to red/gray (maximal) at the left. Black curve shows the final temperature Tf of the
return Loschmidt peak (E ! kBTf hEi) as a function of k.
Logarithm is decimal.
FIG. 4 (color online). Ratio of final to initial temperatures (solid curves) as a function of k, from top to bottom for kBT0=Er 2 106 (blue/black), kBT0=Er 1:8 105
(green/light gray), and kBT0=Er 2 104 (red/gray), with
2 and tr 10. Dashed curves show the theory (3). 䊉
show P2
for kBT0=Er 2 104, confirming theory (4).
Inset: same ratio as a function of for k 4:5; solid curve shows numerical data and dashed curve is the theory (3). PRL 100, 044106 (2008) P H Y S I C A L R E V I E W L E T T E R S 1 FEBRUARY 2008week ending
factor of 100 is reached for 10% of atoms. We also checked that a significant fraction of the atoms returns not only in momentum space but also in coordinate space. To this end we define the fraction of atoms Px which are in the return Loschmidt peak and in the coordinate space interval [2=, 2=]. The data show that Pxis close to the value of P for moderate values of tr 10, so that the Loschmidt cooled atoms remain close both in momentum and coordinate space and thus can be efficiently captured by a trap potential or a Feschbach resonance. We note that at large trvalues P can become smaller than the fidelity
f jh 0j 2trij2, which characterizes proximity of the
whole wave functions and not only the reversed peak. In conclusion, we have presented a concrete experimen-tal proposal of time reversal of matter waves of ultracold atoms in the regime of quantum chaos. If the atoms were classical particles, they would never return back due to exponential instability of dynamical chaos. But the quan-tum dynamics is stable and thus a large fraction of the atoms returns back even if the time reversal is not perfect. This fraction of the atoms exhibits Loschmidt cooling which can decrease their temperature by several orders of magnitude. The reversed peak is very sensitive to varia-tions of and other parameters breaking time reversal symmetry, and therefore this setup can be used as a sensi-tive Loschmidt interferometer to explore such a symmetry breaking (e.g., a gravitational field component along the optical lattice gives a shift in which affects the Loschmidt peak). The parameters considered here are well accessible to nowadays experimental setups (see, e.g., [13–20] ). The realization of our proposal will shed a new light on the long-standing Boltzmann-Loschmidt dispute on time reversibility.
We thank CalMiP for access to their supercomputers and the French ANR (project INFOSYSQQ) and the EC project EuroSQIP for support.
[1] J. Loschmidt, Sitzungsberichte der Akademie der Wissenschaften, Wien, II 73, 128 (1876).
[2] L. Boltzmann, Sitzungsberichte der Akademie der Wissenschaften, Wien, II 75, 67 (1877).
[3] I. P. Kornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory (Springer, New York, 1982).
[4] B. V. Chirikov, Phys. Rep. 52, 263 (1979).
[5] A. Lichtenberg and M. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992).
[6] B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, Sov. Sci. Rev., Sect. C 2, 209 (1981); Physica (Amsterdam) 33D, 77 (1988).
[7] D. L. Shepelyansky, Physica (Amsterdam) 8D, 208 (1983). [8] T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric,
Phys. Rep. 435, 33 (2006).
[9] Pulsed Magnetic Resonance: NMR, ESR, and Optics: A Recognition of E. L. Hahn, edited by D. M. S. Bagguley (Oxford University Press, New York, 1992).
[10] A. Derode, P. Roux, and M. Fink, Phys. Rev. Lett. 75, 4206 (1995); J. de Rosny, A. Tourin, and M. Fink, ibid. 84, 1693 (2000).
[11] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Mon-taldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004); G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[12] S. Fishman, in Quantum Chaos: E. Fermi School Course CXIX, edited by G. Casati, I. Guarneri, and U. Smilansky (North-Holland, Amsterdam, 1993), p. 187.
[13] F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995). [14] H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen,
Phys. Rev. Lett. 80, 4111 (1998).
[15] S. Schlunk, M. B. d’Arcy, S. A. Gardiner, and G. S. Summy, Phys. Rev. Lett. 90, 124102 (2003).
[16] J. Chabe´, G. Lemarie´, B. Gre´maud, D. Delande, P. Szriftgiser, and J. C. Garreau, arXiv:0709.4320. [17] C. Ryu, M. F. Andersen, A. Vaziri, M. B. d’Arcy, J. M.
Grossman, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 96, 160403 (2006).
[18] G. Behinaein, V. Ramareddy, P. Ahmadi, and G. S. Summy, Phys. Rev. Lett. 97, 244101 (2006).
[19] J. F. Kanem, S. Maneshi, M. Partlow, M. Spanner, and A. M. Steinberg, Phys. Rev. Lett. 98, 083004 (2007). [20] M. Sadgrove, M. Horikoshi, T. Sekimura, and K.
Naka-gawa, Phys. Rev. Lett. 99, 043002 (2007).
[21] F. M. Izrailev and D. L. Shepelyanskii, Theor. Math. Phys. 43, 553 (1980).
[22] D. L. Shepelyansky, Physica (Amsterdam) 28D, 103 (1987).
[23] At 0 there is also a significant cooling related to ballistic propagation at the main quantum resonance; however, this effect takes place only in a very narrow range making it rather sensitive to perturbations. FIG. 5 (color online). Fidelity f (䊏), fraction Pof atoms in
the Loschmidt peak (䉱), and fraction Pxof these atoms in the
coordinate space cell (䊉) (see text) as a function of time reversal moment tr. Straight dashed curve shows the fit logP
1:13 logtr 0:43. Parameters are as in Fig. 1. Logarithms
are decimal.
PRL 100, 044106 (2008) P H Y S I C A L R E V I E W L E T T E R S 1 FEBRUARY 2008week ending
